6.6 Systems of Linear Inequalities Word ProblemsUse a system of linear inequalities to model a real-life situation.

Standard Form of Word Problems

•Define variables•Write out the system•Graph the inequalities•Write 2 possible solutions

Word Problem #1You can work a total of no more than 41 hours

each week at your two jobs. Housecleaning pays $5 per hour and your sales job pays $8 per hour. You need to earn at least $254 each week to pay your bills. Write a system of inequalities that shows the various numbers of hours you can work at each job.

x = housecleaning hoursy = sales job hoursHours: x + y ≤ 41Money: 5x + 8y ≥ 254

Can I work negative hours?Be sure to include and

Word Problem #2Fuel x costs $2 per gallon and fuel y costs

$3 per gallon. You have at most $18 to spend on fuel. Write and graph a system of linear inequalities to represent this situation.

x = gallons of fuel xy = gallons of fuel y

Price: 2x + 3y ≤ 18Gallons of x: x ≥0Gallons of y: y ≥ 0

Word Problem #2Graphing…Price: 2x + 3y ≤ 18Gallons of x: x ≥0Gallons of y: y ≥ 0

Think about your intercepts when graphing the first inequality

Word Problem #2Graphing…Price: 2x + 3y ≤ 18Gallons of x: x ≥0Gallons of y: y ≥ 0

The shaded region are All the possible solutionsThat satisfy each inequality

Possible answers( 1,1) (2, 4) (0,6)

Word Problem #3A salad contains ham and chicken. There

are at most 6 pounds of ham and chicken in the salad. Write and graph a system of inequalities to represent this situation.

x = lbs of hamy = lbs of chicken

Total Pounds: x + y ≤ 6Pounds of ham: x ≥ 0Pounds of chicken: y ≥ 0

Word Problem #3Graphing…Total Pounds: x + y ≤ 6Pounds of ham: x ≥ 0Pounds of chicken: y ≥ 0

Remember we cannot have negative lbs!!

Possible solutions1 lb of ham and 1 lb of chicken3 lbs of ham and 2lbs of chicken

Word Problem #4

Mary babysits for $4 per hour. She also works as a tutor for $7 per hour. She is only allowed to work at most 13 hours per week. She wants to make at least $65. Write and graph a system of inequalities to represent this situation.

x = hours babysittingy = hours tutoringHours: x + y ≤ 13Money: 4x + 7y ≥ 65Constraints:

Word Problem #4

Graphing…x + y ≤ 134x + 7y ≥ 65

Word Problem #4

Graphing…x + y ≤ 134x + 7y ≥ 65

Possible Solutions2 hours babysitting and 9 hours tutoring

1 hour babysitting and 10 hours tutoring

Example 5

In one week, Ed can mow at most 9 times and rake at most 7 times. He charges $20 for mowing and $10 for raking. He needs to make more than $125 in one week. Show and describe all the possible combinations of mowing and raking that Ed can do to meet his goal. List two possible combinations.

Earnings per Job ($)

Mowing

Raking

20

10

Example 5- Continued

Step 1 Write a system of inequalities.

Let x represent the number of mowing jobs and y represent the number of raking jobs.

x ≤ 9

y ≤ 7

20x + 10y > 125

He can do at most 9 mowing jobs.

He can do at most 7 raking jobs.

He wants to earn more than $125.

Step 2 Graph the system.

The graph should be in only the first quadrant because the number of jobs cannot be negative.

Solutions

Example 5 Continued

Step 3 Describe all possible combinations. All possible combinations represented by ordered pairs of whole numbers in the solution region will meet Ed’s requirement of mowing, raking, and earning more than $125 in one week. Answers must be whole numbers because he cannot work a portion of a job.

Step 4 List the two possible combinations.Two possible combinations are:

7 mowing and 4 raking jobs 8 mowing and 1 raking jobs

Example 5 - Continued